The Risk Model with Interest,Liquid Reserves and a Constant Dividend Barrier

In this paper, we consider the compoundPoisson surplus model with interest, liquid reserves and a constantdividend barrier. When the surplus of an insurer is below a fixedlevel, the surplus is kept as liquid reserves, which does not earninterest. When the surplus attains the level, the surplus willreceive interest at a constant rate. When the surplus hits anotherfixed higher lever, the excess of the surplus over this higher levelwill be distributed to the shareholders as dividends. We derive asystem of integro-differential equations for the Gerber-Shiudiscounted penalty function and obtain the solutions to theseintegro-differential equations. In the case where the claim sizesare exponential distributed, we get the exact solutions of zerodiscounted Gerber-Shiu function. We also get theintegro-differential equation for the expectation of the discounteddividends until ruin which is the key to discuss the optimaldividend barrier. And we give the exact solution in the special casewith exponential claim sizes.